Finding limits numerically

[Nullvodi]

I'm confused...
Say you have a question such as this one:

	The function values for g(x) near x = 0 are given in the table below. x​ -0.10​ -0.05​ -0.01​ 0.01​ 0.05​ 0.10​ g(x)​ 2.8​ 2.9​ 2.99​ 3.00​ 1.99​ 1.9​ 1.8​ Use the table of values to evaluate
	0 2 does not exist 3

My personal opinion is that the limit does not exist, since you are given the values; the limits should be undefined, yet at x -> 0 the value is clearly 3 (which, by the way, is a wrong answer.

Similarly, this confuses me.

.	Suppose f (x) is a function such that . Which of the following can be true? I. f (5) = 0​ II. f (5) = 2​ III. f (5) is undefined​
	II only I and II only I only I, II, and III Again, I would say that I, II and III are all correct, but I'm not sure. Could someone pleeeeeease help ?Thanks in advance:grin:

 

[mmm4444bot]

Were you told that these exercises concern continuous functions?

 

[HallsofIvy]

IF you are to assume that the limit exists, then the best you can do is take the value at x closest to 0. Is there a reason you have every value of x listed except for the "y= 3.00" term?

 

[DrPhil]

I'm confused...
Say you have a question such as this one:

	The function values for g(x) near x = 0 are given in the table below. x​ -0.10​ -0.05​ -0.01​ 0.01​ 0.05​ 0.10​ g(x)​ 2.8​ 2.9​ 2.99​ 3.00​ 1.99​ 1.9​ 1.8​ Use the table of values to evaluate

My personal opinion is that the limit does not exist, since you are given the values; the limits should be undefined, yet at x -> 0 the value is clearly 3 (which, by the way, is a wrong answer.

In order for a limit to exist, the limit from below must equal the limit from above. The table suggests to me that the limit from below is 3 and the limit from above is 2 [or thereabouts], so the function is discontinuous at x=0. I agree with your personal opinion: The limit does not exist.

 

[DrPhil]

Similarly, this confuses me.

.	Suppose f (x) is a function such that . Which of the following can be true? I. f (5) = 0​ II. f (5) = 2​ III. f (5) is undefined​
	II only I and II only I only I, II, and III Again, I would say that I, II and III are all correct, but I'm not sure. Could someone pleeeeeease help ?Thanks in advance:grin:

Now it really matters whether you know your function is continuous, or whether it is allowed to be discontinuous. What is the answer if continuous? Otherwise? Again, I agree with your answer. Consider this piecewise function:

f(x) = 2 if x /= 5
f(5) = . . .

[I used "/=" for "not equal." Could somebody tell me how to write equations? Thanks.]

 

[HallsofIvy]

In order for a limit to exist, the limit from below must equal the limit from above. The table suggests to me that the limit from below is 3 and the limit from above is 2 [or thereabouts], so the function is discontinuous at x=0. I agree with your personal opinion: The limit does not exist.

I agree completely but the critical word is "suggests". There is no reason why a function, even a continuous one must be "smooth". It is quite possible that the function is getting closer and closer to 2 from above until we get to 0.005 and then suddenly turns up to 3.00. Yes, "not continuous" is probably what is intended but there is no way of being sure from what is given.

 

[JeffM]

Could somebody tell me how to write equations? Thanks.]

Dr. Phil

I presume you know LaTex, which this site supports. Type \(\displaystyle \text{\(\displaystyle Your text here will render in LaTex\)}\).

Two warnings.

(1) The LaTex parser does not give error messages. It simply does not parse if it detects an error. Finding the error is up to you. That makes it really important to use the Preview button to make sure that your LaTex is rendering properly.

(2) You may have noticed that certain words in your posts are colored by the system. An example is function. These represent links to tutorials on the site for those words. If you use one of these words for the first time in your post within LaTex, that line of LaTex will not render when you post, but will render in the preview. The solution is to find a different word to use in your LaTex, or to alter the rest of your post to use that word outside of LaTex first. This little glitch does not come up often, but it is annoying when it does.

I like to use LaTex because I think students benefit from having explanations be as readable and familiar in appearance as possible.

\(\displaystyle \dfrac{(x - 1)(x - 2)^2}{x + 1}\) is less of a reading challenge than (x - 1)(x - 2)^2/(x + 1), and I manage to forget to include sufficient grouping symbols sometimes.

And \(\displaystyle \displaystyle \left(\sum_{i=1}^n\dfrac{P * r}{(1 + r)^i}\right) + \dfrac{P}{(1 + r)^n}\) would be just plain awful in plain text.

Hope this helps.

 

[Bob Brown MSEE]

Problem #1



Note that the limit seems to be different as you approach zero from opposite directions.

Problem #2
I agree with your answer